Context. Galaxies in the local Universe are thought to require ongoing replenishment of their gas reservoir in order to maintain the observed star formation rates. Cosmological simulations predict that such accretion can occur in both a dynamically hot and cold mode, depending on the redshift, halo mass and the details of the included feedback processes. However, until now observational evidence of the accretion required to match the observed star formation histories is lacking. Aims. Within the framework of the Hydrogen Accretion in LOcal GalaxieS (HALOGAS) survey, this paper attempts to determine whether galaxies in the local Universe possess a significant reservoir of cold neutral gas and what would be the accretion rates derived from such reservoirs. Additionally, with this moderately sized sample we can start to investigate whether the observed accretion is connected to intrinsic properties of the galaxies like Hubble type, star formation rate or environment. Methods. We search the vicinity of 22 nearby galaxies for isolated H i clouds or distinct streams, that are not yet connected to the galaxy's disk, in a systematic and automated manner. The HALOGAS observations were carried out with the Westerbork Synthesis Radio Telescope and represent one of the most sensitive and detailed H i surveys to date. These observations typically reach column density sensitivities of ∼ 10 19 cm −2 over a 20 km s −1 width. Results. We find 14 secure H i cloud candidates without an observed optical counterpart in the entire HALOGAS sample. These cloud candidates appear to be analogues to the most massive clouds detected in the extensive cloud distributions around the Milky Way and M 31. However, on average their numbers seem significantly reduced compared to these galaxies. Within the framework of cold accretion, we constrain upper limits for H i accretion in the local Universe. The average H i mass currently observed in a state suggestive of accretion onto the galaxies amounts to a rate of 0.05 M yr −1 with a stringent upper limit of 0.22 M yr −1 , confirming previous estimates. This is much lower than the average star formation rate in this sample. Our best estimate, based on the Green Bank Telescope detection limits of several galaxies in the sample, suggests that at most another 0.04 M yr −1 of neutral hydrogen could be accreted from clouds and streams that remain undetected. Conclusions. These results show that in nearby galaxies neutral hydrogen is not being accreted at the same rate as stars are currently being formed. Our study can not exclude that other forms of gas accretion, such as those provided by direct infall of ionized intergalactic gas or the condensation of coronal gas, triggered by galactic fountain activities, are at work. However, these observations also do not reveal extended neutral gas reservoirs around most nearby spiral galaxies.